Honda's theory gives an explicit description up to strict isomorphism of formal groups over perfect fields of characteristic $p\neq 0$ and over their rings of Witt vectors by means of attaching a certain matrix, which is called its type, to every formal group. A dual notion of right type connected with the reduction of the formal group is introduced while Honda's original type becomes a left type. An analogue of the Dieudonné module is constructed and an equivalence between the categories of formal groups and right modules satisfying certain conditions, similar to the classical anti-equivalence between the categories of formal groups, and left modules satisfying certain conditions is established. As an application, the $\star$-isomorphism classes of the deformations of a formal group over and the action of its automorphism group on these classes are studied.

We show that under certain symmetry, the images of complete harmonic embeddings from the complex plane into the hyperbolic plane is completely determined by the geometric information of the vertical measured foliation and is independent of the horizontal measured foliation of the corresponding Hopf differentials

We establish an analogue of Beurling's uncertainty principle for the group Fourier transform on the Euclidean motion group. We also prove the most general version of Hardy's theorem on it which characterises functions on the motion group that are controlled by the heat kernel associated to the Laplacian of the Euclidean space.

We show that the parabolicity of a manifold is equivalent to the validity of the 'divergence theorem' for some class of $\delta$-subharmonic functions. From this property we can show a certain Liouville property of harmonic maps on parabolic manifolds. Elementary stochastic calculus is used as a main tool.

We shall discuss the local triviality in the ideal class group of the basic $\mathbf Z_p$-extension over an imaginary quadratic field and prove, in particular, a result which implies that such triviality distributes with natural density $1$.

Smooth compact complex surfaces admitting non-trivial surjective endomorphisms are classified up to isomorphism. The algebraic case was dealt with earlier by the authors. The following surfaces are listed in the non-algebraic case: a complex torus, a Kodaira surface, a Hopf surface with at least two curves, a successive blowups of an Inoue surface with curves whose centers are nodes of curves, and an Inoue surface without curves satisfying a rationality condition.

We consider, respectively, the Dirichlet problem and the initial-boundary value problem of elliptic and parabolic equations with two power nonlinearities. We find that these problems are closely related to the so-called quenching problem. We obtain the existence and nonexistence of positive solutions to these problems on bounded and unbounded domains, by using the results of quenching problem and sub-super solution method.